6.1.3 Simulated Fields Visualization and Results Comparison

Now the simulation results are compared with the solutions of the analytical formulas (6.8) and (6.9) for low frequencies and (6.10) and (6.11) for high frequencies. The coaxial structure used is very well suited. It satisfies exactly the homogeneous Neumann boundary conditions.

Table 6.1: Numerically simulated and analytically calculated $R$ and $L$ .

$f$ [GHz]	$\delta$ [$\mu$ m]	$L$ [pH]	$L$ [pH]	$R$ [$\Omega$ ]	$R$ [$\Omega$ ]
		numeric	analytic	numeric	analytic
0.003	47.14	0.664398	0.664112	0.00335153	0.00335063
3	1.491	0.623599	0.639796	0.00479862	0.00387773
30	0.471	0.487201	0.486669	0.0137691	0.0128489
300	0.149	0.43899	0.438291	0.0426505	0.0416856

The following dimensions are used: $a = 3 \mu$ m, $b = 6 \mu$ m, $c = 9 \mu$ m, and $l = 3 \mu$ m. These dimensions are typical for microelectronics applications. For these dimensions skin effect is observable at frequencies over $3$ GHz. Of course it is a question of scaling, because, if larger dimensions are used, the same effects arise at lower frequencies. The outer dielectric layer is needed only for low frequencies, since for high frequencies there is no current density distribution on the outer face of the outer conductor. Then the homogeneous Neumann boundary conditions are also fulfilled without the outer dielectric layer. However, for low frequencies the homogeneous Neumann boundary conditions are fulfilled on the outer shell surface only using the outer dielectric cylinder. Its thickness can be chosen arbitrarily.

Numerical and analytical calculations are performed for different frequencies between $3$ MHz and $300$ GHz. The related results and skin depths are shown in Table 6.1. The field densities distributions in the region of interest are depicted in Fig. <6.4> - Fig. <6.11>. A remarkable short skin depth is observed at $300$ GHz, which is a high frequency for the given dimensions. For $3$ MHz the skin depth is much larger than the thickness of the conductors. For the given structure this is a low frequency case described analytically by (6.8) and (6.9). For this frequency the numerical results agree quite well with the analytical ones. For the remaining frequencies skin effect is observed and the low frequency analytical formulas (6.8) and (6.9) are wrong. In this case (6.10) and (6.11) should be used. However the requirement $d{\ll}b$ , which must be fulfilled for (6.10) and (6.11), is approximately provided only for the highest frequency at $300$ GHz. At this frequency $d$ is set to $3\delta$ in (6.10) and (6.11) and the analytical results match the numerical ones very well. For $3$ GHz and $30$ GHz the condition $d{\ll}b$ is not satisfied and the numerical results differ strongerly from the analytical ones. In such cases the numerical method has to be used.